In this expository talk we relate the spectral properties of a discrete
Schr"odinger operator on a d-dimentional lattice to its dynamical
features. Dynamical quantities of interest include Fourier transforms of
spectral measures, time averaged moments of the position operator, as well
as time-averaged observables for a compact operator. The RAGE theorem in
its various formulations predicts the asymptotic behaviour of these
quantities for any state in the continuous subspace of the Hilbert space:
observables for a compact operator decay to zero, whereas the moments of
the position operator asymptotically diverge. In order to quantify this
decay/divergence, we present a decomposition of the spectral measure with
respect to Hausdorff measures of dimension $\alpha \in [0,1]$. This
decomposition due to Rogers and Taylor generalizes the classical
decomposition of the spectral measure w.r.t. Lebesgue measure into pure
point and continous component. Whereas for $\alpha = 1$ it recasts the
classical result, for $\alpha < 1$ one obtains a decomposition different
to the classical one. For each Hausdorff dimension, the spectral measure
then splits in an $\alpha$-continuous and an $\alpha$-singular component.
$\alpha$-continuous measures are shown to be limits of uniformly $\alpha$
H"older continuous (U$\alpha$H) measures w.r.t. to a suitable topology.
For U$\alpha$H spectral measures lower and upper bounds for various
dynamical quantities are available.
references:
Y. Last, Quantum dynamics and decompositions of singular continuous
spectra, J. Funct. Anal 142, 406-445 (1996).
W. Kirsch: An invitation to random Schr"odinger operators,
arXiv:0709.3707v1[math-ph].
G. Teschl: Mathematical Methods in Quantum mechanics with application to
Schroedinger operators, Graduate Studies in Mathematics, Amer. Math. Soc.,
Providence, 2008. (to appear).
